Algebra in the Stone-\(\check{C}\)ech compactification: applications to topologies on groups

Journal Algebra Discrete Math.

Journal “Algebra and Discrete Mathematics”

Algebra in the Stone-

C

ˇ

ech compactification:

applications to topologies on groups

I. V. Protasov

Communicated by V. I. Sushchansky

Abstract. _{For every discrete groupG, the Stone-Cech com-}ˇ pactification βG of G has a natural structure of compact right topological semigroup. Assume thatG is endowed with some left invariant topologyℑand letτbe the set of all ultrafilters onG con-verging to the unit ofGin ℑ. Thenτ is a closed subsemigroup of βG. We survey the results clarifying the interplays between the al-gebraic properties ofτ and the topological properties of(G,ℑ)and apply these results to solve some open problems in the topological group theory.

The paper consists of 13 sections: Filters on groups, Semigroup of ultrafilters, Ideals, Idempotents, Equations, Continuity in βG andG∗_{, Ramsey-like ultrafilters, Maximality, Refinements,}

Resolv-ability, Potential compactness and ultraranks, Selected open ques-tions.

Introduction

LetGbe a discrete group with the uniteand letβGbe the Stone-Cechˇ compactification ofG. We take the points ofβGto be the ultrafilters on G with the points of G identified with the principal ultrafilters. Using the universal property of βG, we extend the group multiplication on G to the semigroup operation on βG. Formally, the product of ultrafilters pand q is defined by the rule

2000 Mathematics Subject Classification: 22A05, 22A15, 22A20, 05A18, 54A35, 54D80.

Key words and phrases: Stone-Cech compactification, product of ultrafilters,ˇ

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A∈pq⇐⇒ {g∈G: g−1_{A∈q} ∈p.}

With this operation, βG is a right topological semigroup, i.e. the mapping x 7−→ xs is continuous for every s ∈ βG. The semigroup βG is interesting because of its applications to combinatorial number theory and to topological dynamics as well as for its own sake [10].

Now assume that Gis endowed with some left invariant topology ℑ, i.e. all shifts x7−→gx,g∈Gare continuous. Then the set

τ ={p∈βG: pconverges to einℑ}

is a closed subsemigroup ofβG. Thus, we get general problem of study the interplays between the properties of left invariant (in particular, group) topology ℑon Gand the algebraic structure of corresponding semigroup of ultrafiltersτ. The following striking results were obtained on this way.

• There exists in ZFC a maximal regular left topological group (in particular, a maximal regular homogeneous space).

• Every non-discrete left topological group of second category can be partitioned into countably many dense subsets.

• Every countable Abelian groupG with only finite numbers of ele-ments of order 2 can be partitioned into countably many subsets that are dense in every group topology on G.

• If there exists a maximal topological group, then there exists a P -point in ω∗

. If there exists a non-discrete irresolvable topological group which is either Abelian or countable, then there exists a P-point in ω∗_.

• IfG is a countable discrete group and F is a finite subgroup from βG, then there exist a finite subgroup H from G and an idempotent p∈βG such thatF =Hpand p commutes with every element ofH. In particular, every finite subgroup ofβZ _{is a singleton.}

In the first part (sections 1-6) we present the results on algebraic structure of τ which are necessary for applications in the second part (sections 7-12). We conclude the paper with the list of selected open questions (section 13).

1. Filters on groups

Letϕbe filter on a group Gwith the unite,A⊆G. A subset

cl(A, ϕ) ={g∈G: gF ∩A6=∅ for every F ∈ϕ}

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int (A, ϕ) ={g∈G: gF ⊆A for someF ∈ϕ}

is called aninteriorof Awith respect to ϕ.

Letϕ, ψbe filters onG. We define aproductϕ, ψby the rule: A∈ϕ, ψ if and only ifint (A, ψ)∈ϕ. By this definition, the family

{S_g∈F gHg: F ∈ϕ, Hg∈ψfor every g∈F}

forms a base for the filterϕψ. The following basic identity

int (int (A, ψ), ϕ) =int (A, ϕψ)

shows that the product is associative.

A topology ℑ on a group G is called left invariant if the mapping x 7−→ gx is continuous for every g ∈ G. A group G endowed with a left invariant topology is calledleft topological. Since every left invariant topologyℑon Gis uniquely determined by the filterτ of neighborhoods ofe,Gendowed withℑwill be denoted by (G, τ). A filterϕ on a group Gis calledleft topologicalifϕis the filter of neighborhoods ofefor some left invariant topology onG.

For characterization of left topological filters we need some more def-initions. Given an arbitrary filterϕonG, we denote byint (ϕ)the filter on G with the base {int (F, ϕ) : F ∈ ϕ}, and by Clϕ the mapping, determined by the ruleClϕ(A) =cl(A, ϕ) for everyA⊆G.

Theorem 1.1. For every filterϕon a groupG, the following statements are equivalent

(i) ϕ is left topological,

(ii) ϕϕ=ϕ ande∈F for everyF ∈ϕ,

(iii) ϕ=int (ϕ),

(iv) Clϕ is a closure operator.

Theorem 1.2. For every filter ϕon a group G, the filter int (ϕ) is left topological.

Theorem 1.3. For every ultrafilter ϕ on a group G, the left topological group (G, int (ϕ)) is Hausdorff and zero-dimensional. If ϕϕ = ϕ then

(G, int (ϕ)) is extremely disconnected.

For every filter ϕ on a groupG, we denote by hull (ϕ) the maximal (by inclusion) left topological filter onGsuch thathull(ϕ)⊆ϕ.In other words, hull (ϕ) is the filter of neighborhoods of e for the strongest left invariant topology in which ϕconverges toe.

Journal Algebra Discrete Math.

such that {x}SU is a neighborhood of x. Every strongly extremally disconnected space is extremally disconnected.

Theorem 1.4. For every groupG and every ultrafilter ϕ on G, the left topological group (G, hull (ϕ)) is strongly extremally disconnected.

Now letGbe a discrete group,Xbe a topological space and(g, x)7−→

g(x) be a continuous action of G on X. We fix an arbitrary element x0 ∈ X and denote by ϕ the filter on G with the base consisting of

the subsets of the form {g ∈ G : g(x0) ∈ U} where U runs over all

neighborhoods of x₀. It is easy to check that ϕ is left topological. On the other hand, if ϕis a left topological filter onG, thenϕarises in this way from the left action ofG onX = (G, ϕ) withx0 =e.

Comments. For proofs of these statements see [20] and [41]. Every countable group G admits a Hausdorff topology in which all mapping x 7−→ gx, x 7−→ xg, g ∈ G and x 7−→ x−1 _{are continuous. For every}

countable zero-dimensional homogeneous space X and every countable group G, there is a left invariant topology on G such that G is homeo-morphic toX [61].

2. Semigroup of ultrafilters

LetX be a discrete space and letβX be the Stone-Cech extension ofˇ X. We take the point ofβX to be the ultrafilters onX with the points ofX identified with the principal ultrafilters. We putX∗

=βX\X. For every subset A ⊆X, letA¯ ={q ∈βX : A∈ q}. The topology of βX can be defined by stating that the family {A¯: A ⊆X} is a base for the open sets. For every filter ϕon X, the subsetϕ¯=T{A¯: A∈ϕ}is closed in βX, and, for every nonempty closed subset K⊆βX, there exists a filter ϕ on X such that K = ¯ϕ. We put ϕ∗ _{= ¯ϕ}T_X∗_{. Let Y be a compact}

Hausdorff space. For every mappingf : X −→Y, we denote by fβ the Stone-Cech extension ofˇ f onto βX.

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is a right topological semigroups. For the structure of βG and plenty of its combinatorial application we address to the books [10] and [27].

For every left topological filterτ on G,τ andτ∗

are closed subsemi-groups ofβG. The semigroup τ is called thethe semigroup of ultrafilters of the left topological group(G, τ). In what follows we use the semigroups of ultrafilters to explore the topological properties of left topological and topological groups.

LetS be a closed subsemigroup ofβGand letϕbe a filter onGsuch that S = ϕ. We say that S is uniform if ϕ is a left topological filter. Equivalently, S is uniform if, for every U ∈ϕ, there exists V ∈ ϕ, such thatV ϕ⊆U. Every finite subsemigroup ofβGis uniform. For examples of non-uniform subsemigroups see [20] and [11].

By definition, for every left topological group (G, τ), τ is the set of all ultrafilters onGconverging toe. Now we assume that(G, τ)is Haus-dorff and denote by Cτ the set of all converging ultrafilters on (G, τ). Forq ∈Cτ and the limit g of q, we put π1(q) =g, π2(q) = g−1q. Thus,

we have defined the bijection π=π1×π2 between (G, τ) and the direct

product(G, τ)×τ of the left topological group(G, τ)and the right topo-logical semigroupτ. In the following theoremsβGmeans the Stone-Cechˇ compactification of a discrete groupG.

Theorem 2.1. For every Hausdorff left topological group (G, τ), the following statements are equivalent

(i) Cτ is a subsemigroup of βG andπ1 is a homomorphism,

(ii) the mapping x7−→g−1_{xg is continuous at e in (G, τ) for every}

g∈G.

Theorem 2.2. For every Hausdorff left topological group (G, τ), the following statements are equivalent

(i) Cτ is a subsemigroup of βG andπ1, π2 are a homomorphisms,

(ii) for every element g ∈G, there exists U ∈τ such that gx= xg for eachx∈U.

Theorem 2.3. For every Hausdorff left topological group (G, τ), the following statements hold

(i) π₁ is continuous if and only if (G, τ) is regular,

(ii) π2 is continuous if and only if (G, τ) is discrete.

Theorem 2.4. Let (G, τ) be a Hausdorff semitopological group. Then Cτ is a subsemigroup ofβG and the following statements hold

(i) (G, τ) is a homomorphic image of Cτ,

(ii) (G, τ) is a continuous homomorphic image of Cτ provided that

(G, τ) is regular,

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(G, τ) is a continuous homomorphic image of βG. If G is commutative, βG is algebraically isomorphic to G×τ.

Comments. Most results of this section are from [29].

3. Ideals

Every compact Hausdorff right topological semigroupS contains at least one minimal left ideal. Every minimal left ideal is closed and the union M(S)of all minimal left ideals ofSis the minimal ideal ofS. For algebraic structure ofM(S)see [10, Chapter 1] or [47, Chapter 1]. Note that every minimal left ideal L of S is of the form L=Sx for every x∈ L, so the following theorem describes all minimal left ideal of closed subsemigroups of βS.

Theorem 3.1. Let τ be a filter on a group G such thatτ is a subsemi-group ofβG, and let p be an ultrafilter on Gsuch that p∈τ andp∈τ p. Thenτ pis a minimal left ideal ofτ if and only if, for everyA∈pand ev-ery U ∈τ, there exists a finite subset F ⊆U such that cl (F−1_{A, p)∈τ.}

Corollary 1. Let (G, τ) be a left topological group and let p∈τ. Then τ p is a minimal ideal of semigroup τ if and only if, for every A∈p and every neighborhood U of e, there exists a finite subset F ⊆ U such that cl (F−1_{A, p) is a neighborhood of e.}

Corollary 2. Let (G, τ) be a left topological group and let V be a neigh-borhood of the unit e. For every finite partition V = A1S...SAn and every neighborhood U ofe, there existAi and a finite subset F ⊆U such that F−1_A

iA−_i1 is a neighborhood of e.

Corollary 3. For every group G and every finite partition G=A1S...

S_A

n, there exist Ai and a finite subset F of G such that G=F AiA−i1 and AiA−_i 1AiA_i−1∈p for every p∈βG such that pp=p.

If L is a minimal left ideal of βG, thenG acts continuously onL by the rulex7−→gx,x∈L,g∈G, and Lis the minimal universalG-space (see [18] and [53]).

Now we define some ideals and describe some ideal decomposition of the semigroup of ultrafilters of left topological group.

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is called an o-ultrafilter. By Zorn Lemma, every o-filter is contained in some o-ultrafilter, but o-ultrafilter needs not to be an ultrafilter.

Ifϕis a filter on(G, τ),τ ⊆ϕandϕτ =ϕ, we say thatϕis a uniform right ideal of the semigroupτ .It is easy to check thatϕis a uniform right ideal if and only ifϕis an o-filter.

Given an arbitrary filter ϕon(G, τ)such thatτ ⊆ϕ, there exists the maximal o-filter o(ϕ) satisfying o(ϕ) ⊆ϕ, which is called the open hull of ϕ. Clearly, ϕ is an o-filter if and only if ϕ = o(ϕ). Moreover, ϕ is an o-filter if and only if ϕ= o(p) for every ultrafilter p on G such that ϕ⊆p.

An ultrafilter p∈τ is calledpreopen if int(F, τ)∈p for every closed in(G, τ) subset F ∈p. An ultrafilterp∈τ is preopen if and only ifo(p)

is an o-ultrafilter. The set of all preopen ultrafilters from τ forms the closed ideal ofτ.

Theorem 3.2. For every left topological group (G, τ), the ideal of all preopen ultrafilters is disjoint union of the uniform right ideals ϕ, where ϕruns over all o-ultrafilters from τ.

We say that a filter ϕ ⊆ τ is a c-filter if ϕ has a base consisting of the subsets ofG\ {e}that are closed inG\ {e}. A filter that is maximal in the class of all c-filters is called c-ultrafilter. By Zorn Lemma, every c-filter is contained in some c-ultrafilter, but c-ultrafilter needs not to be an ultrafilter.

Theorem 3.3. Let (G, τ) be a nondiscrete Hausdorff left topological group such that the space G\ {e} is normal. Then the semigroup τ∗ _is

disjoint union of the uniform right ideals o(ϕ), where ϕ runs over all c-ultrafilters on(G, τ).

A c-filter ϕ on (G, τ) is called primitiveif, for any closed in G\ {e}

disjoint subsets A, B,ASB ∈ ϕimplies either A ∈ϕ or B ∈ϕ. Every c-ultrafilter is primitive.

Theorem 3.4. For every nondiscrete Hausdorff left topological group

(G, τ), the semigroup τ∗ _{is a union of the uniform right ideals o(ϕ),}

where ϕruns over all primitive ultrafilters from τ.

Note that the union in Theorem 3.3 could not be disjoint.

The applications of the above notions are based on the following ob-servations.

• A left topological group (G, τ) is extremally disconnected if and only ifτ has only one uniform right ideal.

Journal Algebra Discrete Math.

• A left topological group (G, τ) is irresolvable (=Gcan not be par-titioned onto two dense subsets) if and only if every preopen ultrafilter is an o-ultrafilter.

Comments. The results of this section are from [19] and [21].

4. Idempotents

An elementpof a semigroupS is called an idempotent ifpp=p. Denote byE(S)the set of all idempotents ofS and define three natural preoders on E(S)

e≤l f if and only ife=ef, e≤r f if and only if e=f e, e≤f if and only ife=ef =f e.

Every compact right topological semigroup has an idempotent [10, Theorem 2.5]. In particular, for every discrete group G, every closed subsemigroup ofβGhas an idempotent. For combinatorial and dynamical applications of this theorem see [10] and [18], [53].

Every idempotent p∈βGdetermines two topologies(G, int(p))and

(G, hull p) which will be characterized in Section 9. Here we define two special types of idempotents ofβG (namely, strongly right maximal and strongly summable) of great importance for constructions of extremal topologies on group.

By [10, Theorem 2.12], every compact right topological semigroup has a right maximal idempotent, i.e. an idempotent, which is maximal with respect to≤r. IfGis a countable discrete group andpis a right maximal idempotent in βG, then the subsemigroup {q ∈ βG : qp = p} is finite [10, Theorem 9.4].

LetGbe an infinite discrete Abelian group,p∈G∗_{be an idempotent.}

Then, for every subset P ∈p, there exists an infinite subset A⊆P such that F S(A) ⊆ P, where F S(A) is the set of all finite sums of distinct elements of A. This observation easily implies the Hindman’s Theorem: for every finite partitionG=A1S...SAn, there exist Ai and an infinite subset A⊆Ai such thatF S(A)⊆Ai.

An idempotentp∈G∗ _{is calledstrongly summableif, for everyP ∈p,}

there exists an infinite subsetA⊆P such thatF S(A)⊆P andF S(A)∈

p. By [12, Theorem 2.8], Martin’s Axiom (MA) implies that there is a strongly summable ultrafilter p ∈G∗_{. On the other hand [12, Theorem}

3.6], the existence of strongly summable ultrafilters on G implies the existence of P-point in ω∗

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An element r of a semigroup S is called right cancellable if, for any p, q ∈ S, pr = qr implies p = q. Let G be a countable discrete group, r ∈βG. By [10, Theorem 8.18], p is right cancellable in βG if and only if there is no idempotentp∈G∗

such thatpr=r.

5. Morphisms

A topological spaceX with the distinguished element eX (the identity) and a partial binary operation (the multiplication) is called a local left topological groupif there is a left topological group Gsuch that

• eX is the identity ofG,

• X is an open neighborhood of eX inG,

• the partial multiplication on X is precisely the partial operation induced onX by the multiplication on G.

Note that every left topological group is a local left topological group. Let X, Y be local left topological groups. A mapping f : X −→

Y is called a local homomorphism if f(eX) = eY and, for every x ∈ X, there exists a neighborhood U of eX such that, for all y ∈ U, the productsxy,f(x)f(y)are defined andf(xy) =f(x)f(y). If, in addition, f is a homeomorphism, then f is called alocal isomorphism. If the left topological groups (G, τ) and (G′_{, τ}′

) are locally isomorphic, then the semigroupsτ and τ′ _{are topologically isomorphic.}

Theorem 5.1Any two nondiscrete countable Hausdorff zero-dimensional local left topological groups of countable weight are locally isomorphic.

A local automorphism f of a local left topological group X is called a homogeneous local automorphism of order m (m is a natural number) if the f-orbit O(f, x) = {fn(x) : n < ω} of every element x ∈ X, x 6= eX is of cardinality m. We consider the following example. Let

Z_{m+1={0,1, ..., m}be the cyclic group and let H(m+ 1) =⊕ω}Z_{m+1 be}

the direct sum ofωdiscrete copies ofZ_{m+1 endowed with the topology of}

pointwise convergence, soH(m+ 1)is a topological group. Letsmbe the coordinate-wise substitution on H(m) induced by a substitution onZ_m+1

which is the product of independent cycles(1)(2,3, ..., m+ 1). Clearly, f is a homogeneous local automorphism of orderm.

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X −→h H(m)

f ↓ sm ↓

X −→h H(m)

Theorem 5.3Let G, H be countable discrete groups and let ϕ: βG−→

H∗

be a continuous homomorphism. Thenϕ(βG) is a finite group.

Theorem 5.4Let Gbe a countable discrete groups,H be a finite discrete group and let ϕ: G∗ _−→H

be a continuous homomorphism. Then there exists a homomorphismf : G−→H such thatϕ=fβ_|

G∗.

Theorem 5.5. LetG, H be countable discrete groups,ϕ: G∗_−→H∗

be a continuous surjective homomorphism. Then there exists a homomor-phism f : G−→H such that ϕ=fβ|G∗.

Comments. The notion of local left topological semigroup was intro-duced by E. Zelenyuk. Theorems 5.1 and 5.2 are parts of more general Theorem 3.1 from [60]. Originally, Theorem 5.1 arouse as the positive an-swer to the author’s question: are the semigroups of ultrafilters of any two countable metrizable topological group topologically isomorphic? The most remarkable application of local left topological groups is Zelenyuk’s Theorem, stating that every finite subgroup of βGis a singleton, where G is a countable discrete torsion-free group. For proof of this theorem see [57] or [10, Chapter 7]. The finite subgroups of βG, where G is a countable discrete group, were characterized in [31]. These all have the formF p, whereF is a finite subgroup ofGandpis an idempotent which commutes with the elements of F.

For the case G=H =Z_{, Theorem 5.3 was proved by D.Strauss [52]}

as the answer to question of van Douwen. Theorems 5.3, 5.4 and 5.5 are from [38]. For discontinues automorphisms see [29].

6. Equations

Some problems concerning joint continuity in βG (see Section 7) or re-solvability of left topological groups (see Section 11) can be reduced to the equations in βG. Here we present some statements about corresponding equations.

Let Gbe an Abelian group, p∈βG. For every integerm, by mpwe denote the ultrafilter onG with the base{mP : P ∈p}.

Theorem 6.1. Let G be an infinite discrete Abelian group, p ∈ G∗_,

s∈ Z_{, s6= 0,1. If the subgroup {g ∈G: s(s−1)g = 0} is finite, then}

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s∈Z_{, s6= 0,1. Suppose that one of the following condition holds}

(i) there exists P ∈ p such that every element of P has an infinite order,

(ii)for everyP ∈p, the set of all prime divisors of orders of elements of P is infinite.

Then the equation x+p=y+sp is not solvable in βG.

In other words, these theorems states that, under corresponding con-ditions,βG+pTβG+sp=∅.

Theorem 6.3. If Gis a discrete Abelian group, p∈βG andp+p= 2p, then p is a principal ultrafilter .

Theorem 6.4. For every p ∈ Z∗

and every integer s, s 6= 0,1, the equation p+x=sp+y is not solvable in βZ_.

It follows from Theorem 6.4 that, for every idempotent p ∈ Z∗ _and

every integers, s6= 0,1,the subgroup of βG generated by p and spis a free product of the subsemigroups{p}and {sp}.

Comments. These results are from [23] and [24]. For generalizations see [13].

7. Continuity in βG and G∗

Let G be a discrete group, p, q ∈ βG. We say that (p, q) is a point of joint continuityif the multiplicationβG×βG−→βGis continuous at the point(p, q). It is easy to see that(p, q)is a point of joint continuity if and only if, for every R ∈p, q, there exist P ∈p, Q∈q such that P Q⊆R. If either p∈G or q ∈G, then (p, q) is a point of joint continuity. These points of joint continuity are calledstandard.

Let Gbe a countable discrete Abelian group, p, q∈G∗

. If (p, q) is a point of joint continuity, then there existsr∈G∗ _{such that the equation}

x+r=y+2rhas a solution inβG. This reduction from [23] and Theorem 6.1 give us

Theorem 7.1. If G is a countable discrete Abelian group with finite subsets of elements of order 2, then every point of joint continuity in βG×βG is standard.

LetG be a discrete group. For everyp∈βG, we define the mapping λp : βG−→βGby the rule λp(q) =pq. The set

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is called the topological center ofβG. For every p ∈G∗_{, we denote by λ}∗

p the restriction of λp to G∗. The set

Λ(G∗

) ={p∈G∗

: λ∗

p is continuous at every point q∈G∗}

is called the topological center ofG∗_.

Theorem 7.2. For every discrete groupG, Λ(G∗_{) =∅. Moreover, if |G|}

is nonmeasurable, then there exists q∈G∗

such that λ∗

p is discontinuous at every point q∈G∗_.

CorollaryFor every discrete group G,Λ(βG) =G.

Theorem 7.3. Suppose that a countable discrete G is isomorphic to some subgroup of compact topological group. If q ∈ G∗ _{and, for every}

p∈G∗_{, λ}∗

p is continuous atq, then q is aP-point in G∗.

Theorem 7.4. Let G be a countable discrete group and let p ∈ G∗

be either an idempotent or a right cancellable element of βG. If λ∗

p is continuous at p, then there exists a mapping f : G −→ ω such that fβ(p) is a P-point in ω∗

.

Given an arbitrary discrete group G, we denote by µ: βG−→ βG the mapping defined by µ(q) =qq. Let µ∗ _{be the restriction of µto G}∗_.

Theorem 7.5. Let G be a countable discrete group and let p ∈ G∗

be either an idempotent or a right cancellable element of βG. If µ∗ _is

continuous at p, then there exists a mapping f : G −→ ω such that fβ(p) is a P-point in ω∗

.

Theorem 7.6. Let G be a countable discrete Abelian group with finite set of elements of order 2. Ifp∈G∗ _andµ∗ _{is continuous at p, thenp is}

a P-point in G∗

.

In spite of above theorems, the mappingsλ∗

p andµ∗ have some conti-nuity-like property.

Let X, Y be topological spaces. A mapping f : X −→ Y is called quasi-continuous at point x ∈ X if, for every neighborhoods U and V of x and f(x), there exists a nonempty open subset W ⊆ U, such that f(W)⊆U.

Theorem 7.7. Let G be a discrete group and let p ∈ G∗_{. Then µ}∗ _is

quasi-continuous at p. If G is countable, then λ∗

p is quasi-continuous at p.

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Theorem 7.2 is from [28], Corollary from Theorem 7.2 and the first statement of this theorem in the case of countable groups is due to [9]. For generalization of Theorem 7.2 see [40].

The remaining theorems of this section are from [45].

8. Ramsey-like ultrafilters

It follows easily from the Ramsey theorem that, for every coloring χ :

G −→ {0,1} of an infinite Abelian group G, there exists an infinite subset A ⊆G such that the subset P S(A) ={a+b: a, b ∈A, a 6=b}

is monochrome. We say that an ultrafilter p ∈G∗

is a P S-ultrafilterif, for every coloring χ: G−→ {0,1}, there existsA∈p such that P S(A)

is monochrome. Clearly, every Ramsey (=selective) ultrafilter on Gis a P S-ultrafilter. An Abelian group G is called a Boolean group if2g = 0

for every g ∈ G. Every strongly summable ultrafilter P (see Section 4) on an infinite Boolean group is aP S-ultrafilter but p is not selective. If p is a P S-idempotent on a countable Boolean group G, then (G, int p)

is a maximal topological group (see Section 1).

A subset Aof an Abelian group Gis called 2-independentor a Sidon set if, for any two-elements subsets{a, b},{c, d}ofA,a+b=c+dimplies

{a, b} ={c, d}. A P S-ultrafilter is selective if and only if there exists a 2-independent subsetA∈p.

Theorem 8.1. EveryP S-ultrafilter on an infinite Abelian group without elements of order 2 is selective.

Theorem 8.2. Let G be an infinite Abelian group, p ∈G∗_{, H = {g ∈}

G : 2n_{g = 0 for some n ∈ N}. If p is a P S-ultrafilter then either p is} selective or there exists g∈Gsuch that g+H∈p.

An ultrafilterpon a setXis called aQ-point, if, for every partition of X onto finite subsets, there existsP ∈p such that|PTA| ≤1 for every cellA of the partition. It follows from Theorem 8.2 that ifP S-ultrafilter pis a Q-point thenp is selective.

Theorem 8.3. Let G be an infinite Abelian group, p be a P S-ultrafilter on G such that, for every P ∈ p, there exist a, b ∈ P, a 6= b such that a+b∈P. Then p+p=p and 2p= 0.

Theorem 8.4. If p is a P S-ultrafilter on a countable Abelian group G, then either p is right cancellable or there exist p′ _∈G∗_{, g∈ G such that}

p=g+p′

,p′

+p′

=p′

.

Theorem 8.5. Let Gbe a countable Abelian group,p be aP S-ultrafilter on G. Then there exists a mapping f : G −→ ω such that fβ(p) is a P-point inω∗

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Let H be a subgroup of an infinite Abelian group G. We say that an ultrafilterp ∈G∗ _{is H-selectiveif either there exists g∈G such that}

g+H ∈ p or there exists P ∈ p such that |PT(g+H)| ≤ 1 for every g∈G. We say thatpissubselectiveifpisH-selective for every subgroup H ofG. EveryP S-ultrafilter is subselective, but the class of subselective ultrafilters is much more wider than the class ofP S-ultrafilters.

Theorem 8.6. For every infinite Abelian group G, there exists a sub-selective ultrafilter p ∈ G∗

such that p+p = p and kpk = |G| where

kpk=min{|P|: P ∈p}.

Theorem 8.7. Every free ultrafilter on an infinite Abelian group G is subselective if and only if every infinite subgroupHofGhas a finite index in G.

Comments. TheP S-ultrafilters were introduced in [25], all results of this section are from [39].

9. Maximality

All topological spaces in this section are suppose to be Hausdorff. A topological space X is called maximal if X has no isolated points but X has an isolated points in every stronger topology. We say that a left topological group is maximal if it is maximal as a topological space.

Let G be an infinite discrete group with the identity e. For every idempotent p ∈ G∗_{, we put τ}

p = {PS{e} : P ∈ p}. Then τp is a left topological filter and the left topological group(G, τp)is maximal. In the notations of Section 1, τp = hull(p) so τp is the strongest left invariant topology onGin whichp converges toe. In what follows, we writeG(p)

instead of (G, τp).

On the other hand, if (G, τ) is a maximal left topological group, then τ∗

={p},p is an idempotent of G∗

and (G, τ) =G(p). Thus, there is a bijection between the set off all maximal left invariant topologies on G and the set of all idempotents of G∗_{. It follows that, for every infinite}

groupG, there are 22|G| maximal left invariant topologies on G.

For every filter ϕ on a set X, we put kϕk = min{|F|: F ∈ ϕ}. If

(G, τ) is a nondiscrete left topological group, then there exists an idem-potentp∈τ∗_{such thatkpk=kτk. In other words, the topology of(G, τ)}

can be strengthened to the maximal left invariant topology (G, τ′

) such thatkτ′_k=kτk.

As distinct from the topological group case, a left invariant topology on a group needs not to be regular. Given an idempotent p∈G∗_{, let ϕ}

Journal Algebra Discrete Math.

left invariant topology onGin whichpconverges toe. In the notations of Section 1, it can be shown thatϕp =int(p) soϕp ={q ∈βG: qp=p}. In what follows we writeG[p]instead of(G, int p). By Theorem 1.3,G[p]

is zero-dimensional and extremally disconnected.

Theorem 9.1. Let(G, τ)be a countable nondiscrete regular left topolog-ical group of countable weight. Then there exists an idempotent p ∈ τ∗

such that G[p] =G(p).

Corollary. Every infinite group G admits a regular maximal left in-variant topology. In particular, there exists (in ZFC) a regular maximal homogeneous space.

The proof of Theorem 9.1 uses the right maximal idempotent and Theorem 5.1.

Theorem 9.2. Every maximal left topological group is a union of a countable family of closed discrete subspaces.

It follows from Theorem 9.2 that every maximal left topological group has countable pseudocharacter, i.e. the identity eofGis an intersection of some countable family of open subsets. Equivalently, every countable complete idempotent fromβGis a principal ultrafilter.

Let(G, τ)be an infinite left topological group. The cardinalκis called theindex of boundednessof (G, τ)if κ is the minimal cardinal such that, for everyU ∈τ, there existsF ⊆G such that G=F U and |F|< κ. If the index of boundedness of(G, τ) isℵ0,(G, τ)is called totally bounded.

Theorem 9.3. The index of boundedness of every maximal left topolog-ical group(G, τ) is greater than the cofinality of |G|.

Theorem 9.4. Let G be an infinite group,p be an idempotent from the minimal ideal of the semigroup βG. Then G[p] is totally bounded.

It follows from Theorem 9.4, that every infinite group admits a zero-dimensional totally bounded left invariant topology.

By [ 43, Theorem 2], for every free group G and every idempotent p ∈G∗

, the cellularity of the left topological group G[p] is |G|. In view of Theorem 9.4, there are the totally bounded left topological groups of arbitrarily large cellularity.

Now we present some results concerning the maximal topological groups.

Theorem 9.5. LetGbe an infinite group,p∈G∗_{. IfG[p]is a topological}

group, thenG[p]contains a countable open Boolean subgroup.

If(G, τ)is a maximal topological group, then(G, τ) =G(p) for some idempotent p ∈ G∗

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G(p) =G[p]. Hence, Theorem 9.5 is a generalization of Malykhin’s theo-rem [16]: every maximal topological group has a countable open Boolean subgroup.

Under MA, an example of maximal topological group was constructed by Malykhin in [15].

Let G be a countable Boolean group, p be an idempotent from G∗_.

Then G(p) is a topological group if and only if p is a PS-ultrafilter (see Section 8). In particular, every strongly summable ultrafilter on G de-termines the maximal group topology. It was shown in [20] that the existence of a maximal topological group implies the existence ofP-point inω∗

. The following theorem is a generalization of this statement.

Theorem 9.6. Let Gbe a group, p be an idempotent fromG∗ _{such that}

G[p] is a topological group. Then there exists a mapping f : G −→ ω such that fβ(p) is a P-point in ω∗

.

Theorem 9.7. Every maximal topological group is complete in the left uniformity.

It is interesting that if there exists a maximal topological group then there exists a maximal topological group with distinct left and right uni-formities.

A groupGprovided with a topology is calledsemitopologicalifGis left and right topological. A semitopological group is called quasitopological if the mapping x 7−→ x−1 _{is continuous. A group G provided with a}

topology is called paratopological if the multiplication (x, y) −→ xy is jointly continuous.

Theorem 9.8. For every group Gprovided with a topology, the following statements are equivalent

(i) Gis a maximal semitopological group,

(ii)Gis a maximal left topological group and, for every elementg∈G, there exists a neighborhood U of the identity such that xg=gx for every x∈U.

Theorem 9.9. For every group Gprovided with a topology, the following statements are equivalent

(i) Gis a maximal quasitopological group,

(ii) Gis a maximal semitopological group and there exists a neighbor-hoodU of the identity esuch that x2 =e for every x∈U.

Corollary. LetGbe a maximal quasitopological group. Then every neigh-borhood of the identity contains an infinite Boolean subgroup.

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infinite Boolean group and let p be an idempotent fromG∗

. ThenG(p)

is the maximal quasitopological group. In view of Corollary from Theo-rem 9.9 it is naturally to ask (see [34] or [3 , Question 3)]: does there exist an open Abelian subgroup in every maximal quasitopological group? The negative answer (in ZFC) to this question was done in [44].

Theorem 9.10. Every maximal paratopological group is a topological group.

Comments. All results of this section are from [33] and [34]. For proof of Theorem 9.1 see also [10, Chapter 9]. The topologies on semigroups determined by idempotents were considered in [11].

10. Refinements

As we have seen in the previous section, maximality is a very exotic phe-nomenon in the category of topological groups because there exist ZFC-models without maximal topological groups. In contrast to topological groups every nondiscrete left topological group has the left topological refinements that are maximal. But from the topological point of view these refinements could be very far from the original left invariant topol-ogy on a group. In this section we consider some new kinds of refinements of the left invariant topologies. On one hand, these refinements are very close to the original topology. On the other hand, they have a wide spec-trum of extremal topological properties. The results presented in this section witness that the category of left topological groups is much more appropriate environment for life of extremal objects than the category of topological groups. Moreover, the extremal objects are not exotic in this category because they can be constructed from any left topological groups in some very natural ways.

Letτ, ϕbe left topological filters on a groupG. We say that(G, ϕ)is anopen refinementof(G, τ) ifτ ⊆ϕand every nonempty open subset of

(G, ϕ) contains a non-empty open subset of(G, τ). If in additionτ 6=ϕ,

(G, ϕ) is called a proper open refinement of (G, τ). A left topological group (G, τ) is called o-maximal if it has no proper open refinements. By Zorn Lemma, for every left topological group, there exists a maxi-mal open refinement. Every maximaxi-mal open refinement is o-maximaxi-mal. To characterize these refinements we use the o-ultrafilters from Section 3.

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{e}SU is a neighborhood of e.

It follows from above theorems that every left topological group has a strongly extremally disconnected (see Section 1) open refinement. We note also that some basic cardinal invariants of left topological groups (density, cellularity, index of boundedness) are stable under the open refinements.

Let τ, ϕ be left topological filters on a group G, τ ⊆ ϕ. We say that (G, ϕ) is a dense refinement of (G, τ) if cl (U, τ) ∈ τ for every U ∈ ϕ. If in addition τ 6=ϕ, (G, ϕ) is called a proper dense refinement. A left topological group is called d-maximal if it has no proper dense refinement. By Zorn Lemma, every left topological group has a maximal dense refinement and every maximal dense refinement isd-maximal.

Let (G, τ) be a left topological group and let ϕbe a filter on Gsuch that τ ⊆ ϕ. We put (τ ϕ)e = {FS{e} : F ∈ τ ϕ} where e is the identity of G. Then (G,(τ ϕ)e) is a dense refinement of (G, τ). If (G, τ) is d-maximal, then(τ p)e =τ for every ultrafilter p ∈ τ. It follows that τ =LS{e} whereLis the minimal left ideal of τ.

Let τ, ϕ be left topological filters on a group G, τ ⊆ϕ. We say that

(G, ϕ) is anopen-dense refinement of(G, τ) if it is both open and dense refinement. If in addition τ 6= ϕ, (G, ϕ) is called a proper open-dense refinement of(G, τ). A left topological group is called od-maximal if it has no proper open-dense refinements.

Given any left topological filterτ on a groupG, consider the strongest filterψ onGcontaining all o-filters on(G, τ). We callψ theopen coreof τ. Clearly, ψis the strongest filter containing all o-ultrafilters on (G, τ), so p ∈ψ if and only if p is preopen in (G, τ). By Theorem 1.1, ψe is a left topological filter. For every subset A ⊆ G, we have A ∈ ψ if and only if int (ATU, τ) =6 ∅ for every open subset U in (G, τ) such that e∈cl(U, τ).

Theorem 10.3. Let(G, τ)be a left topological group and letψbe an open core of τ. Then(G, ψe) is the unique maximal open-dense refinement of

(G, τ).

A left topological group (G, τ)is called nodecif every nowhere dense subset in (G, τ) is closed. Note that every nowhere dense subset of a nodec group is discrete.

Theorem 10.4. For every left topological group (G, τ), the following statements are equivalent

Journal Algebra Discrete Math.

(ii) every ultrafilter p∈τ∗

is preopen,

(iii) (G, τ) is od-maximal.

By Theorems 10.3 and 10.4, every left topological group has an open-dense refinement which is nodec.

Theorem 10.5. A nondiscrete left topological group is maximal if and only if it is o-maximal and nodec.

Now we give an ultrafilter characterization of extremally disconnected left topological groups.

Theorem 10.6. Let (G, τ) be a left topological group and let ϕ be an o-ultrafilter on (G, τ). Then the left topological group (G, int ϕ) is zero-dimensional and extremally disconnected.

Theorem 10.7. Let (G, τ) be a left topological group and let ϕ be an o-ultrafilter on (G, τ). Then (G, τ) is extremally disconnected if and only if int(ϕ)⊆τ. If (G, τ) is regular, then (G, τ) is extremally disconnected if and only ifint (ϕ) =τ.

By Theorem 10.6 and 10.7, every regular left topological group has a zero-dimensional extremally disconnected open refinements.

A regular left topological group(G, ϕ) is called abounded refinement of a regular left topological group (G, τ) if τ ⊆ϕ and, for every subset U ∈ ϕ, there exists a finite subset K ⊆ G such that KU ∈ τ. If in additionτ 6=ϕ,(G, ϕ)is called a proper bounded refinement of(G, τ).We say that(G, τ)isb-maximalif it has no proper bounded refinements. By Zorn Lemma, every regular left topological group has a maximal bounded refinement. Clearly, every maximal bounded refinement is b-maximal.

Theorem 10.8. A regular left topological group (G, τ) is a maximal bounded refinement of a regular left topological group(G, τ)if and only if there exists an idempotent p from the minimal ideal of the semigroup τ such thatϕ=int(p). A regular left topological group(G, τ)is b-maximal if and only if ϕ=int (p) for every idempotent p from the minimal ideal of τ.

By Theorem 1.3, for every infinite group G and every idempotent p ∈ G∗

, (G, int (p)) is zero-dimensional and extremally disconnected. Applying Theorem 10.8, we conclude that every regular left topologi-cal group admits a zero-dimensional extremally disconnected bounded refinement.

Theorem 10.9. Let(G, τ)be a regular nondiscrete left topological group. A left topological group(G, ϕ)is a maximal regular nondiscrete refinement of (G, τ) if and only if there exists a right maximal idempotent p of the semigroup τ∗

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By Theorem 1.4, for every group G and every ultrafilterp on G, the left topological group(G, hull(p))is strongly extremally disconnected.

Theorem 10.10. Let G be an infinite Boolean group and let p be a Ramsey ultrafilter on G. Then the left topological group (G, int(p)) is a topological group.

Given any group G and any p, q ∈ βG, we say that p, q are locally isomorphic if the corresponding left topological group(G, hull(p))and

(G, hull (q))are locally isomorphic (see Section 5). The problem of clas-sification of ultrafilter on a group up to the local isomorphism is very difficult. The next theorem gives a solution in one special case.

Theorem 10.11. Let G be a countable group and let p, q be the right cancellable ultrafilters from G∗_{. Then p, q are locally isomorphic if and}

only if there exists a bijection f : G−→Gsuch that fβ(p) =q.

We note that a free ultrafilter p on a countable group G is right cancellable if and only if there exists a subsetP ∈psuch that the identity ofGis the only limit point ofpis(G, hull(p)). For every right cancellable p ∈ G∗_{, the left topological group (G, hull (p)) is Hausdorff and}

zero-dimensional.

Comments. All results of this section are from [41]. It is still un-known: does there exist in ZFC a nondiscrete nodec topological group or a nondiscrete extremally disconnected topological group? Under MA, ev-ery countable Abelian group admits a nondiscrete nodec group topology (see [58] or [35, Chapter 2]). In the case of countable groups, the sec-ond question is an old problem of Arhangel’ski˘i[2]. Under CH, the first example of nondiscrete countable extremally disconnected group was con-structed in [51]. Theorem 10.10 is a generalization of the Sirota’s results. Some partial results incline us to the negative answer to the Arhangel’ski˘i question. Thus, Theorem 9.6 can be re-formulated as follows. If there exists a countable nondiscrete extremally disconnected topological group G such that the topology of G is maximal in the class of all regular left invariant topologies on G, then there is a P-point in ω∗_{. Another}

statement of this type was proved in [59]. If there exists a countable nondiscrete extremally disconnected topological group which has a non-closed discrete subsets then there exists aP-point ofω∗_{. For some partial}

cases of Theorem 10.11 see [54].

11. Resolvability

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X isresolvable. IfX is notκ-resolvable, we say thatX isκ-irresolvable.

Theorem 11.1. Every nondiscrete left topological group of second cate-gory is ℵ0-resolvable.

The following construction is crucial for the proof of Theorem 11.1. LetGbe an uncountable group of cardinalityγ with the unite. Using a minimal well-ordering ofGwe can construct inductively the chain{Gα : α < γ}of subgroups ofG such that

(i) G0 ={e}, G=S{Gα: α < γ},

(ii) Gα⊂Gβ for any α < β < γ,

(iii)S{Gα: α < γ}=Gβ for every limit ordinal β < γ,

(iv) |Gα|< γ for every α < γ.

For every ordinalα < γ, we decomposeGα+1\Gαinto the right cosets by the subgroupGα and fix some subsetXα of representatives of cosets. Thus Gα+1\Gα = GαXα. Take arbitrary element g ∈ G, g 6= e and choose the smallest subgroupGα such thatg∈Gα. By(iii),α=α1+ 1

for some ordinalα1 < γ. Theng∈Gα1+1\Gα1 andg=g1xα1,g1∈Gα1.

Ifg1 6=e, we chooseα2,g2 ∈Gα2,xα2 ∈Xα2 such thatg1 =g2xα2. After

the finite number s(g) of steps we get the representation

g=xα_s(g) xα_s(g)−1 ... xα2 xα1,

α_s(g)< α_s(g)−1 < ... < α2 < α1, xαi ∈Xαi.

Using this representation we define the partition

G\{e}= [

n∈ω Sn+1

whereSn+1={g∈G\ {e}: s(g) =n+ 1}.

Now let (G, τ) be a nondiscrete left topological group, ϕ be an o-ultrafilter on (G, τ). Then (G, τ) is ℵ0-irresolvable if and only if ϕ is

finite. By means of this criterion, the general case of Theorem 11.1 can be reduced to the case in which every subsetSn is dense.

Theorem 11.2. Every infinite groupG can be partitioned into |G| sub-sets dense in every left invariant left totally bounded topology on G.

Theorem 11.3. If (G, τ) is a nondiscrete ℵ0-irresolvable topological Abelian group, then the subgroup {g ∈ G : 2g = 0} is countable and open.

Theorem 11.4. Let(G, τ)be a nondiscrete left topological Abelian group, m be integer such that the mapping x 7−→ mx is continuous and the subgroup {g ∈G : m(m−1)g= 0} is finite. If |m|>1, then (G, τ) is

Journal Algebra Discrete Math.

and ℵ0-irresolvable, then the subgroup{g∈G: 2g= 0} is open.

Theorem 11.6 Let G be a countable group with only finite numbers of elements of order 2. Assume thatGadmits a totally bounded group topol-ogy. Then G can be partitioned into countable many subsets dense in every non-discrete group topology on G.

Theorem 11.7.Let (G, τ) be a non-discrete ℵ0-irresolvable topological group. If G is either Abelian or countable, then there exists a P-point in ω∗_.

Comments. The concept of resolvability of topological space was in-troduced by E.Hewitt in 1944. Theorem 11.1 is from [37] and [42]. It answers the following question posed by A.V.Arhangel’skiˇiand P.Collins [4]: is every submaximal topological group of first category? The ex-amples of non-discrete irresolvable topological spaces of second category under some set-theoretical assumptions are given in [49] and [50].

The start for study resolvability of topological groups was done by W.W.Comfort and J. van Mill [6]: every non-discrete topological Abelian group with only finite set of elements of order 2 is resolvable. Theorem 11.3 sharpens this result and was proved in [26]. Theorem 11.2 is from [17], Theorem 11.4 was proved in [36] by means of equations in βG.

Theorem 1.5 and 1.6 were proved by E.Zelenyuk [60] by means of automorphisms of local left topological groups. Under MA Zelenyuk con-structed an irresolvable topological group which is not maximal [55].

For Abelian groups, Theorem 11.7 was proved in [37], the countable case is reduced to the Abelian case by means of Theorem 11.5.

More on resolvability of topological groups can be found in the surveys [7], [8] and [30]. For combinatorial aspects of resolvability see [5] and [46].

12. Potential compactness and ultra-rank

A group G is called potentially compact if, for every ultrafilter p on G, there exists a Hausdorff group topology onGwith respect to whichp con-verges to some point ofG. Clearly, every group which admits a compact group topology is potentially compact.

Theorem 12.1.Every potentially compact group is a subgroup of some compact topological group.

Journal Algebra Discrete Math.

Applying this criterion we get the following statements.

Theorem 12.3.A free Abelian group Gis potentially compact if and only if rankG >1.

Theorem 12.4.An Abelian torsion groupGis potentially compact if and only if it is reduced and the number of its non-trivial p-Sylow subgroups is finite.

The concept of potential compactness can be generalized via the fol-lowing cardinal invariant of group.

Let G be a group. A family ℑ of ultrafilters on G is called suitable if there exists a Hausdorff group topology on G in which all ultrafilters from ℑ converge. In particular, G admits a compact group topology if and only if βG is suitable. If G does not admit the compact group topologies, we denote byinc(G) the minimal cardinality of non-suitable families of ultrafilters onG.

Let κ be a cardinal and let G be a subgroup of a topological group H. We say that G is topologically κ-servant in H if, for every family

{hα : α∈J}of elements of H such that|J|< κthere exists the family

{gα : α ∈ J} of elements of G such that GT<< X >> = {e}, where X={gαhα: α∈J},<< X >>is the smallest closed invariant subgroup ofH containingX.

Theorem 12.5. If G is a topologically κ-servant subgroup of compact topological group H, then inc(G)≥κ.

Theorem 12.6. Assume that a group G is a subgroup of some com-pact topological group. If G is not topologically κ-servant in the Bohr’s compactificationG♯ of discrete group G, then inc(G)< κ.

Applying Theorem 12.5 and 12.6 we get the following statements.

Theorem 12.7. Every free group of rank>0is not potentially compact.

Theorem 12.8. If inc(G)>ℵ0, then Gis algebraically compact.

Journal Algebra Discrete Math.

To calculate the ultra-rank of topological group we use the preodering on the set of all group topologies on G defined by the rule: τ1 ≤ τ2 if

and only if, for every V ∈ τ1, there exists a finite subset F ⊆ G such

that F V ∈ τ2. It is easy to check that τ1 ≤ τ2 if and only if every

Cauchy ultrafilter in (G, τ2) is a Cauchy ultrafilter in(G, τ1). For every

topological group G there exists the unique maximal topology τ0 such

that τ0 ≤ τ. We say that (G, τ0) is the totally bounded modification of

(G, τ).

Let(G, τ)be a topological group and let(G, τ0)be the totally bounded

modification of (G, τ). We consider G, the completion ofb (G, τ0) in

the two-sided group uniformity and extend the identity mapping f : (G, τ0) −→ (G, τ) to the continuous homomorphism fb: Gb −→ (G, τb ).

We put Fτ =Ker fband say that the subset X ⊆Fτ topologically gen-eratesFτ as the normal divisor ofGb ifFτ is the smallest closed invariant subgroup ofGb containing X.

Theorem 12.9. Let(G, τ) be a topological group. Then u-rank (G, τ) is the minimal cardinality of subsets of Fτ which topologically generate Fτ as the normal divisor of Gb.

Applying Theorem 12.9 we conclude that ultra-rank of every infinite compact Abelian group G is 22|G|. On the other hand, ultra-rank of Z

endowed with the topology of finite indices is 1.

Comments. All results of this section are from [19] and [62]. The preodering on the set of group topologies was defined independently in [19] and [48].

13. Selected open questions

Question 1. Has every maximal topological group a base of neighborhoods of the unit consisting of subgroups? Equivalently, is every PS-ultrafilter on the countable Boolean group strongly summable?

Question 2. LetGbe a countable Abelian group,q be a PS-ultrafilter on G. Is it true that q is either selective or q =a+p for some a∈ G and some idempotent p∈G∗

?

Question 3. Does there exist in ZFC a maximal regular homogeneous spaceX such that every non-empty open subset of X is uncountable?

Question 4. Let X be a non-discrete homogeneous space of second cat-egory. Is X resolvable? ℵ0 - resolvable?

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If the answer to this question is positive then, by Theorem 11.7, it is impossible to construct a non-discrete irresolvable topological group in ZFC.

Question 7. Let G be an infinite Abelian group, m be integer, |m|>1. Assume that the subgroup {g ∈ G : m(m −1)g = 0} is finite. Is it possible to decompose G into countably many subsets that are dense in any topology with continuous shifts and continuous mapping x7−→mx.

Question 8 (V. Malykhin). Let (G, τ) be a countable non-discrete irre-solvable topological group. Is τ¯ finite?

Question 9(I. Guran). LetG, τ be an extremally disconnected paratopo-logical group. Is (G, τ) a topological group?

Question 10. LetGbe a countable discrete group, q∈G∗

. Suppose that the mappingλ∗

p : G

∗

−→G∗

,λ∗

p(x) =pxis continuous at the point q for every p∈G∗_{. Is q a P-point in G}∗_.

Question 11. Does there exist (in ZFC) a countable group Gandp, q∈

G∗ _{such that λ}∗

p is continuous atq?

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Contact information

I. V. Protasov Department of Cybernetics, Kyiv National University, Volodimirska 64, Kyiv 01033, Ukraine

E-Mail: [email protected]